Lesson 28a - Approximating Taylor Series

Lesson 28a - Approximating Taylor Series - (x T f,a x n 0 f...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
Estimating Taylor Series Consider a Taylor polynomial of a function f(x): ) ( , ) )( ( ) ( n n a f a x a f x T Then we find that the approximation stopping after n = k will be given by: (somewhat similar to that of the alternating series test) 0 ! n n ) )( ( ) ( 1 ) 1 ( a x z f x R k k k Where z is some (not usually known) number between a and x depending on how dit i fd h t th t ) i th id f )! 1 ( k many derivatives were performed. In here, we note that R k (x) is the remainder of how close f(x) is to the Taylor expansion stopping after k terms. Proving this theorem is a bit challenging, so we will not do so, but if we can place a reasonable bound on z, we can show that the error is
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Estimating Taylor Series So when does f(x) = T(x)? Consider the following Theorem: Consider a Taylor Series of a function f(x): ) ( , ) )( ( ) ( n n a f a x a f x T en T(x) = f(x) if and only if T(x) x) = 0, if and only if 0 ! n n Then T(x) f(x) if and only if T(x) f(x) 0, if and only if 0 ) ( ) ( lim x T x f k k 0 ) ( lim x R k k 0 ) )( ( lim 1 ) 1 ( a x z f k k )! 1 ( k k
Background image of page 2
Example 1
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 7

Lesson 28a - Approximating Taylor Series - (x T f,a x n 0 f...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online